Do all functions have an output value for any input value?

Question

With the exception of a select few output values, like a division by zero or some other undefined value, do all functions have a value?

Ignore for a second the Cartesian graph... ignore the fact that most definitions of a function require you the ability to plot a value for a given x...

What Im asking is, even if the solution isnt REAL and cant be plotted... does it exist, even if as a complex number?

for example...
Does arcsin (2) exist as a complex number? What value of theta will sin θ yield 2

Do functions exist that dont have -any- value output for a given input, even if only in theory?

Answer 1

Sure they do.

This function:
f(x) = 1/ (-min( 0, -x))

Arcsin(2) has several possible values, usually the one chosen is with real part = pi/2, imaginary part = -1.316958

arcsin(z) = -i*ln[ iz + (1-z^2)^(1/2) ].

Answer 2

In the complex plane
sin z=1/2i *(e^iz-e^-iz) and the equation sin z =k has always a complex solution for z.(call e^iz =t)
Ii z = IzI the function ln z = ln IzI +i(t+2npi) where lnIzI is the real log. and
the imaginary part is multivalued

Answer 3

considering that 0 is the 1st enter am taking -a million and -2 to be the 2d and third respectively if z=0 then f(z)=02-a million+2i=-a million+2i if z=-a million then f(z)=-12-a million+2i=2i is z=-2 then f(z)=-22-a million+2i=4-a million+2i=3+2i determination D you will get it till you make the main of an imaginary style i sq. root of -500i=+-22.36